Bond Pierce
08/26/2024 · Elementary School
\( m ^ { 6 } - 216 n ^ { a } \)
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Step-by-step Solution
The expression \( m^6 - 216n^a \) can be factored or simplified depending on the context or the values of \( a \).
First, notice that \( 216 \) can be expressed as \( 6^3 \), so we can rewrite the expression as:
\[
m^6 - (6n^{a/3})^3
\]
This resembles the difference of cubes, which can be factored using the formula:
\[
x^3 - y^3 = (x - y)(x^2 + xy + y^2)
\]
In our case, let \( x = m^2 \) and \( y = 6n^{a/3} \). Then we can apply the difference of cubes factorization:
\[
m^6 - 216n^a = (m^2 - 6n^{a/3})(m^4 + 6m^2n^{a/3} + 36n^{2a/3})
\]
So, the factored form of the expression is:
\[
(m^2 - 6n^{a/3})(m^4 + 6m^2n^{a/3} + 36n^{2a/3})
\]
This is a complete factorization of the expression \( m^6 - 216n^a \) under the assumption that \( a \) is a multiple of 3. If \( a \) is not a multiple of 3, the expression cannot be factored in a similar manner without additional context or constraints.
Quick Answer
The expression \( m^6 - 216n^a \) can be factored as \( (m^2 - 6n^{a/3})(m^4 + 6m^2n^{a/3} + 36n^{2a/3}) \) if \( a \) is a multiple of 3.
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